Proper names for colleges and universities are of three main types, syntactically. The first, which I’ll call the XU type (for simplicity I limit discussion here to names with the head noun University) has a modifier preceding the head noun, as in Harvard University. The second, the UX type, has a postnominal complement, usually a preposition phrase headed by the preposition of and almost always specifying a location, as in the University of California (UC). The third, the the XUY type, has both prenominal modifier and postnominal complement, as in the City University of New York (CUNY).
The X in an XU name is sometimes the name of a founder or benefactor, and sometimes a location specification as in New York University (NYU) and Indiana University (IU). In a UX name the complement is nearly always a location specifier (the University of the Arts is a rare exception).
The XU/UX contrast may be the sole distinction between two university names: Washington University (in St Louis, Mo.) has a different name from the University of Washington (in Seattle,Wash.); York University (in Toronto, Canada) has a different name from the University of York (in Yorkshire, England).
In Britain, however, universities that officially have UX names are informally referred to by XU variants: Cambridge University and University of Cambridge both get tens of millions of Google hits, far too many for the former to be classed as an error (though officially it’s the University of Cambridge). Thus “York University” will be used informally for the University of York.
In North America things seem stricter: No one seems to refer to the Canadian institution as the University of York, UC is never called *California University (I’ll mark ill-formed variants with an asterisk), NYU is never referred to as *the University of New York, IU is not correctly called *the University of Indiana, and Harvard University is definitely not *the University of Harvard (though I have heard that error from a BBC radio announcer).
Exceptionally, a number of UX universities in the Rocky Mountains and southern Midwest have recognized XU nicknames: The University of Colorado is known as CU, and similarly for Denver (DU), Kansas (KU), Missouri (MU), Nebraska (NU), Oklahoma (OU), and Tulsa (TU).
UC, which stresses that it is a single university, assigns its campuses names like the University of California, Berkeley. Strictly speaking, *the University of California at Berkeley is not correct (though it is house style for some newspapers, including this one). The Wisconsin system uses a dash as separator for campus names (the University of Wisconsin–Madison); Illinois uses the preposition at (the University of Illinois at Urbana-Champaign).
Some campuses of the California State University follow the UC practice (e.g. the California State University, Monterey Bay), but others have adopted XU names identifying them as if they were entirely separate universities (e.g. San Jose State University).
SUNY (see the campus list has XU names for some of its component institutions (Binghamton University) and UX names for others (the University at Albany).
This is just the barest beginning of a grammar of university names, i.e. of one very small subset of the proper names in English. The topic is peripheral to the structure of English, getting barely a mention in The Cambridge Grammar of the English Language’s roughly 1,700 pages of description (Page 516 has a brief remark or two within a four-page treatment of proper names).
Yet if you’re an American academic you probably were already tacitly aware of just about all of my generalizations. If you can read and understand English you have somehow acquired a grasp of a large set of complex and often irregular syntactic patterns that is thousands or even tens of thousands of times more complex than that.
And that raises a question: How did you come by that knowledge? Those of a linguistic nativist persuasion often claim that you cannot learn anything about what is not allowed in a language solely from positive input. That claim forms part of their case that most of our command of language must spring from innate universal grammatical principles. But the patterns of currently accepted university names surely couldn’t owe anything significant to universal grammar.
If you know that both Oxford University and Brown University sound natural, and the University of Oxford does too, but *the University of Brown does not, then you learned at least some fiddly details of English, including some details about variants that are incorrect, through mere exposure to the positive data of other people’s language use. Believe it or not, in the theory of language acquisition that’s a controversial claim.